Do you mean in the Android Settings, you turned Location off? Map has it's own LocationSensor as part of the component (users cannot control it, it just is. You cannot turn on or off that feature). Perhaps Evan will explain.
I ran your aia Will; I see you do not connect a LS. I looked at Patryks video. Your data toy shows activity indicating something. Did you look and test the non Map component OSM access discussion I provided using your data logger? What are the results?
You might continue using the Map with all its foibles, I expect it will be a while before MIT finds the issue. I don't see a serious issue based on my tests but sympatize with you guys. The issue might be with App Inventor, it might be as Patryk suggests an issue with the OSM api accessing their server. Only MIT can figure this out.
In the meanwhile, you might use the Google Maps api along with a WebViewer and do your dataflow check. Google's map tiles used this way may consume less data. Developers can do the same stuff Map can ... zoom, plot locations, plot traks, determine bearings etc. Google Static Map API
Why not try this and see how data flow compares?
Also, continue to post your theories, they may help Evan debug if necessary.